Solve |x + 6| + 3 = 18
Absolute value equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = −21, x = 9 |
Step-by-step solution
8 steps-
1 Given\left|{x + 6}\right| + 3 = 18
Solve for x.
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2 Isolate the absolute value\left|{x + 6}\right| = 15
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3 Split into two cases: the expression inside the bars is either 15 or -15
|u| is the distance of u from 0. A distance of 15 can be on either side of 0, so the inside is 15 or the opposite of 15, and each case is solved as its own equation.
x + 6 = 15\quad \text{or} \quad x + 6 = -15 -
4 Case 1x + 6 = 15
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5 Subtract 6 from both sides
An equation stays true as long as you do the same thing to both sides, like adding the same weight to both pans of a balance. Adding or subtracting the same amount on both sides moves a term to the other side, where it appears with the opposite sign.
x = 9Move the constant terms to the right.
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6 Case 2x + 6 = -15
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7 Subtract 6 from both sidesx = -21
Move the constant terms to the right.
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8 Check\begin{aligned}x = -21:\quad 18 = 18\quad\checkmark\\ x = 9:\quad 18 = 18\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| −21 | 18 | 18 | ✓ |
| 9 | 18 | 18 | ✓ |
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